Now we introduce the weighted Sobolev spaces taken from
[1]} and
[2]}. Take a nonnegative real-valued function \(\zeta (x)=\zeta (x^1)\in C^{\infty }_0(\mathbb {R}_+)\)  such that
\(\sum _{n=-\infty }^{\infty }\zeta (e^{n+s})>c>0, \quad \forall s\in \mathbb {R},\) 
